Chapter 14 – Partial Derivatives

1. Chapter 14 – Partial Derivatives 14.6 Directional Derivatives and the Gradient Vector • Objectives: • Determine the directional derivative of a vector • Determine the gradient of a vector 14.6 Directional Derivatives and the Gradient Vector

2. Definition – Directional Derivative 14.6 Directional Derivatives and the Gradient Vector

3. Visualization • Directional Derivatives 14.6 Directional Derivatives and the Gradient Vector

4. Theorem 14.6 Directional Derivatives and the Gradient Vector

5. Note • If the unit vector u makes an angle  with the positive x-axis, then we can write u = <cos, sin > and the formula in theorem 3 becomes 14.6 Directional Derivatives and the Gradient Vector

6. Example 1- pg. 920 # 4 • Find the directional derivative of f at the given point in the direction indicated by the angle θ. 14.6 Directional Derivatives and the Gradient Vector

7. Example 2 • Find the directional derivative of f at the given point in the direction indicated by the angle θ. 14.6 Directional Derivatives and the Gradient Vector

8. Definition - Gradient • The gradient of f,∇f, is read “del f”. 14.6 Directional Derivatives and the Gradient Vector

9. Direction Derivative and Gradient • We can now rewrite the directional derivative as which expresses the directional derivative in the direction of u as the scalar projection of the gradient vector onto u. 14.6 Directional Derivatives and the Gradient Vector

10. Example 3 – pg. 943 # 8 • Find the gradient of f. • Evaluate the gradient at the point P. • Find the rate of change of f at P in the direction of vector u. 14.6 Directional Derivatives and the Gradient Vector

11. Definition – Directional Derivative • If we use vector notation, then the definition is more compact. 14.6 Directional Derivatives and the Gradient Vector

12. Definition - Gradient • The gradient of f is and can be rewritten as 14.6 Directional Derivatives and the Gradient Vector

13. Example 4 • Find the gradient of f. • Evaluate the gradient at the point P. • Find the rate of change of f at P in the direction of vector u. 14.6 Directional Derivatives and the Gradient Vector

14. Maximizing the Directional Derivative • Visualization • Maximizing the Directional Derivative 14.6 Directional Derivatives and the Gradient Vector

15. Example 5 • Find the maximum rate of change of f at the given point and the direction in which it occurs. 14.6 Directional Derivatives and the Gradient Vector

16. Example 6 • Find the maximum rate of change of f at the given point and the direction in which it occurs. 14.6 Directional Derivatives and the Gradient Vector

17. The equation ∇F(x0,y0,z0)∙r’(t0) says that the gradient vector at P, ∇F(x0,y0,z0),is perpendicular to the tangent vector r’(t0) to any curve C on S that passes through P. 14.6 Directional Derivatives and the Gradient Vector

18. Definition • If ∇F(x0,y0,z0)≠0, it is thus natural to define the tangent plane to the level surface F(x, y, z) = k at P(x0, y0, z0) as the plane that passes through P and has normal vector ∇F(x0,y0,z0). This equation of a tangent plane can be written as 14.6 Directional Derivatives and the Gradient Vector

19. Normal Line • The normal lineto S at P is the line: • Passing through P • Perpendicular to the tangent plane • The direction of the normal line is given by the gradient vector ∇F(x0,y0,z0) and its symmetric equations are 14.6 Directional Derivatives and the Gradient Vector

20. Example 7 • Find the equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 14.6 Directional Derivatives and the Gradient Vector

21. Group work • 1. Pg. 944 # 28 • Find the directions in which the directional derivative of the function below at the point (0,2) has the value of 1. 14.6 Directional Derivatives and the Gradient Vector

22. Group work • 2. Pg. 944# 32 • The temperature at a point (x, y, z) is given by • Where T is measured in oC and x, y, z in meters. • Find the rate of change of temperature at the point P(2,-1,2) in the direction towards the point (3, -3, 3). • In which direction does the temperature increase the fastest at P? • Find the maximum rate of increase at P. 14.6 Directional Derivatives and the Gradient Vector

23. Group work • 3. Pg. 944 # 34 • Suppose you are climbing a hill whose shape is given by the equation , where x, y, and z are measured in meters, and you are standing at the point with coordinates (60, 40, 996). The positive x-axis points east and the positive y-axis points another. • If you walk due south, will you start to ascend or descend? At what rate? • If you walk northwest, will you start to ascend or descend? At what rate? • In which direction is the slope the largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin? 14.6 Directional Derivatives and the Gradient Vector

24. Group work • 4. Pg. 945 # 50 • If find the gradient vector and use it to find the tangent line to the level curve g(x, y)=1 at the point (1,2). Sketch the level curve, the tangent line, and the gradient vector. 14.6 Directional Derivatives and the Gradient Vector

25. More Examples The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 4 • Example 5 • Example 8 14.6 Directional Derivatives and the Gradient Vector

26. Demonstrations • Feel free to explore these demonstrations below. • Directional Derivatives in 3D • Maximizing Directional Derivatives 14.6 Directional Derivatives and the Gradient Vector